Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 29K (Stochastic Financial Models)

The price process is a geometric Brownian motion withPutand define the equivalent measure $Q$ on ${\mathcal{F}}_T$ by the exponential Brownian martingaleThe Cameron-Martin-Girsanov theorem says that $W_t^Q=B_t+\vartheta t$ is Brownian motion under $Q$. Thereforeso $e^{-rt}{S}_t$ is a $Q$-martingale. This is the unique equivalent martingale measure, since cancellation of the single Brownian risk forces the displayed drift shift.

By risk-neutral valuation,Writing $W_T^Q=\sqrt{T}Y$ with $Y\sim N(0,1)$ gives

Let $\tau =T-t$. Conditional on ${\mathcal{F}}_t$ under the risk-neutral measure,Using the moment-generating function of a normal distribution in the risk-neutral expectation gives the Power payoff in the Black-Scholes model valueIn particular,The delta hedge holdsunits of the risky asset.